Functional equation that models trigonometric identities

Question

Find, with proof, all continuous functions $f,g : \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x)^2 + g(x)^2 = 1$ and $2f(x)g(x)=f(2x)$. I am aware that the solution pair $(f,g)=(\sin{ax},\cos{ax})$ satisfies the conditions, but cannot determine what other solutions exist.